If log y = tan^–1 X, show that : (1+x²)y₂+(2x–1) y₁=0.

Note: y₂ represents second order derivative i.e. and y₁ = dy/dx

Given,

log y = tan^–1 X

∴ y = ……equation 1

to prove : (1+x²)y₂+(2x–1)y₁=0

We notice a second order derivative in the expression to be proved so first take the step to find the second order derivative.

Let’s find

As

So, lets first find dy/dx

Using chain rule, we will differentiate the above expression

Let t = tan^–1 x => []

And y = e^t

…….equation 2

Again differentiating with respect to x applying product rule:

Using chain rule we will differentiate the above expression-

[using & ]

Using equation 2 :

∴ (1+x²)y₂+(2x–1)y₁=0 ……proved